An introduction to ferromagnetism

1
An introduction to ferromagnetism

• Freshman Seminar
• Nanoscience and Nanotechnology
• January 31, 2006

2
Origin of magnetic behavior
Orbital motion of charged particles
Flow of charge gives rise to magnetic
field Charges flow in response to magnetic field
Note that particles moving in circular orbits
have angular momentum---orbital angular momentum
Spin angular momentum
Spin is a purely quantum mechanical property Can
think of a particle turning on its own axis
3
Three types of magnetic behavior
Diamagnetic
Lenzs law a magnetic field will induce a
current whose direction is such that it will
create a magnetic field that opposes the change
that produced it. If you apply an external
magnetic field to a diamagnetic material, the
induced magnetization of the material will oppose
the field producing it.
4
Three types of magnetic behavior
Paramagnetic
Consider a magnetic dipole (think small bar
magnet) in a magnetic field
Interaction energy between dipole and magnetic
field is given by
B
?
To minimize energy, dipole moment points along
field Magnetization points along magnetic field
5
Three types of magnetic behavior
Paramagnetic
For both diamagnetic and paramagnetic materials,
there is no magnetization in the absence of an
external field
Example paramagnetic crystal
B
No field, dipoles point in random directions No
net magnetization
Finite field, dipoles point in the same
direction Finite net magnetization
6
Three types of magnetic behavior
Ferromagnetism
For a paramagnet, the interactions between spin
moments are negligible Various types of
interactions (typically mediated by electrons)
may lead to alignment of adjacent spin
moments If the interactions prefer parallel
alignment, then the interaction is said to be
ferromagnetic
If the interactions prefer anti- parallel
alignment, then the interaction is said to
be antiferromagnetic (more later).
J
Note there is no external field yet!
7
Energy scales
Paramagnets
Does the fact that interactions between magnetic
moments in a paramagnet is weak mean that all
moments line up exactly in even a small external
magnetic field? No. While the external field
would like to align them, finite temperature
likes to randomize their directions Alignment of
all magnetic moments depends on the balance
between the energy associated with a moment in
magnetic field, and the thermal energy If
the thermal energy is greater, then the moments
will be randomized Emagneticlt kBT If the
magnetic energy is greater, then the moments will
be aligned Emagneticlt kBT One can write
this as M (some numerical constants)x ? B where
the susceptibility ? is a measure of the magnetic
response and varies with temperature as 1/T
(Curie Law)
8
Energy scales
Ferromagnets
For a ferromagnet, one has similar
behavior However, instead of the magnetic energy
of a magnetic moment in an external magnetic
field, the relevant energy that fights against
temperature is the interaction energy between
moments, J If JltkBT, then the moments will be
randomized--no net magnetization If JgtkBT, then
the moments will order spontaneously, forming a
ferromagnet
Ferromagnetic transition
A ferromagnetic transition is an example of a
phase transition
M
T
Curie temperature
9
Other magnetic systems
Antiferromagnets
If the interactions are antiferromagnet, the
situation is similar to that of a ferromagnet,
except that the moments will alternate (up-down).
The material will have magnetic order, but no
net magnetization. This is called an
antiferromagnet. A similar material is a
ferrimagnet, where the moments alternate, but the
magnitude of each adjacent moment is not equal,
hence the material has a small net moment.
Frustrated magnetic systems
In a system with antiferromagnetic interactions,
the crystal structure may be such that the system
cannot order antiferromagnetically (triangular
lattices, Kagome lattices)
?
10
Ferromagnets
Even a ferromagnet below the Curie temperature
may not have a net magnetization. This is
because the ferromagnet breaks up into domains,
each with its magnetization pointing in a
different direction. One requires a magnetic
field to the moments of all the domains in one
direction.
11
Ferromagnets
As one applies a magnetic field, it progressively
aligns the direction of the magnetization of the
domains. The magnetization of the sample
saturates when all the domains are aligned.
Since moving domains around costs energy, the
magnetization of a ferromagnet is not a
single-valued function of the external fieldthe
magnetization depends on the history of the
sample as well as the external field.
saturation magnetization
The resulting M vs. H curve is called a
hysteresis loop. It costs energy to cycle the
external field through a loop.
M
H
Coercive field
12
Ferromagnets
Domain walls Between two magnetic domains, the
average magnetization must switch direction, as
shown below
The area over which the magnetization reverses is
called a domain wall. The domain wall width is
typically determined by the properties of the
material. Typical domain wall widths for
conventional ferromagnets are of the order of
20-100 nm.
13
Nanoferromagnets
What happens if the dimensions of a ferromagnet
are smaller than a domain wall width? The
particle is then comprised of a single magnetic
domain! Which way will the magnetization
point? A number of factors control the direction
of the magnetization Two are important
here For some materials, the magnetization
prefers to point along specific crystalline
directions (crystalline anisotropy) The shape
also determines the direction due to surface
magnetic energy (shape anisotropy)
14
Single Particle Magnetoresistance
Magnetic field rotates magnetization
Single discrete switch in MRcorresponds
to reversal of magnetization single-domain
particle
Resistance depends on angle between current and
magnetization
Aumentado and Chandrasekhar, (1999)
15
Nanoferromagnets
However, if we make the dimensions of the
nanomagnets too small, thermal fluctuations will
destroy the magnetization (superparamagnetic
limit) This determines the minimum size of the
domains that can be used to store
information This can also be controlled by
controlling material parameters (using different
ferromagnets)
Single domain tracks in a hard disk
16
Micromagnetic simulations
We can simulate the response of a nanomagnetic
particle by integrating the magnetic equations of
motion
500 nm long axis permalloy particle
17
Micromagnetic simulations
We can simulate the response of a nanomagnetic
particle by integrating the magnetic equations of
motion
250 nm long axis permalloy particle
18
Micromagnetic simulations
Using lithographic techniques, we can design
samples to nucleate domain walls in specific
locations